Peer-to-Peer Networks 03 CAN (Content Addressable Network) - - PowerPoint PPT Presentation

Peer-to-Peer Networks

03 CAN (Content Addressable Network)

Christian Ortolf

Technical Faculty Computer-Networks and Telematics University of Freiburg

CAN Playground

• Index entries

are mapped to the square [0,1]2

• using two hash

functions to the real numbers

• according to

the search key

• Assumption:
• hash functions

behave a like a random mapping

2

CAN Index Entries

• Index entries are mapped to

the square [0,1[^2

• using two hash functions to

the real numbers

• according to the search key
• Assumption:
• hash functions behave a like

a random mapping

• Literature
• Ratnasamy, S., Francis, P.,

Handley, M., Karp, R., Shenker, S.: A scalable content- addressable network. In: Computer Communication

• Review. Volume 31., Dept. of
• Elec. Eng. and Comp. Sci.,

University of California, Berkeley (2001) 161–172

Dick Karp Mark Handley Sylvia Ratnasamy Paul Francis Scott Shenker

3

First Peer in CAN

• In the beginning

there is one peer

• wning the whole

square

• All data is

assigned to the (green) peer

4

CAN: The 2nd Peer Arrives

• The new peer

chooses a random point in the square

• or uses a hash

function applied to the peers Internet address

• The peer looks up

the owner of the point

• and contacts the
• wner

5

CAN: 2nd Peer Has Settled Down

• The new peer

chooses a random point in the square

• or uses a hash function

applied to the peers Internet address

• The peer looks up the
• wner of the point
• and contacts the owner
• The original owner

divides his rectangle in the middle and shares the data with the new peer

6

3rd Peer

7

CAN: 3rd Peer

8

CAN: 4th Peer

9

CAN: 4th Peer Added

10

CAN: 5th Peer

11

CAN: All Peers Added

12

On the Size of a Peer‘s Area

• R(p): rectangle of

peer p

• A(p): area of the

R(p)

• n: number of peers
• area of playground

square: 1

• Lemma
• For all peers we have
• Lemma
• Let PR,n denote the

probability that no peers falls into an area R. Then we have

13

An Area Not being Hit

• Lemma
• Let PR,n denote the probability that

no peers falls into an area R. Then

• Proof
• Let x=Vol(R)
• The probability that a peer does

not fall into R is

• The probability that n peers do not

fall into R is

• So, the probability is bounded by
• because

R

14

How Fair Are the Data Balanced

• Lemma
• With probability n-c a rectangle of size

(c ln n)/n is not further divided

• Proof
• Let PR,n denote the probability that no

peers falls into an area R. Then we have

• Every peer receives at most

c (ln n) m/n elements

• if all m elements are stored equally

distributed over the area

• While the average peer stores

m/n elements

• So, the number of data stored on

a peer is bounded by c (ln n) times the average amount

15

Network Structure of CAN

• Let d be the dimension
• f the square, cube,

hyper-cube

• 1: line
• 2: square
• 3: cube
• 4: ...
• Peers connect
• if the areas of peers share

a (d-1)-dimensional area

• e.g. for d=2 if the

rectangles touch by more than a point

16

Lookup in CAN

• Compute the position of

the index using the hash function on the key value

• Forward lookup to the

neighbored peer which is closer to the index

• Expected number of

hops for CAN in d dimensions:

• Average degree of a

node

• 17

Insertions in CAN = Random Tree

• Random Tree
• new leaves are inserted

randomly

• if node is internal then flip

coin to forward to left or right sub-tree

• if node is leaf then insert

two leafs to this node

• Depth of Tree
• in the expectation: O(log n)
• Depth O(log n) with high

probability, i.e. 1-n-c

• Observation
• CAN inserts new peers like

leafs in a random tree

18

Leaving Peers in CAN

• If a peer leafs
• he does not announce it
• Neighbors continue testing on the

neighborhood

• to find out whether peer has left
• the first neighboir who finds a missing neighbor

takes over the area of the missing peer

• Peers can be responsible for many

rectangles

• Repeated insertions and deletions of peers

leed to fragmentation

19

Defragmentation — The Simple Case

• To heal fragmented areas
• from time to to time areas are

freshly assigned

• Every peer with at least two

zones

• erases smalles zone
• finds replacement peer for this

zone

• 1st case: neighboring zone

is undivided

• both peers are leafs in the

random tree

• transfer zone to the neighbor

20

Defragmentation — The Difficult Case

• Every peer with at least two

zones

• erases smalles zone
• finds replacement peer for this

zone

• 2nd case: neighboring zone

is further divided

• Perform DFS (depth first

search) in neighbor tree until two neighbored leafs are found

• Transfer the zone to one leaf

which gives up his zone

• Choose the other leaf to

receive the latter zone

21

DFS

Improvements for CAN

• More dimensions
• Multiples realities
• Distance metric for routing
• Overloading of zones
• Multiple hasing
• Topology adapted network construction
• Fairer partitioning
• Caching, replication and hot-spot management

22

Higher Dimensions

• Let d be the

dimension of the square, cube, hyper- cube

• 1: line
• 2: square
• 3: cube
• 4: ...
• The expected

path length is O(n1/d)

• Average

number of neighbors O(d)

23

More Realities

• Build simultanously

r CANs with the same peers

• Each CAN is called

a reality

• For lookup
• greedily jump

between realities

• choose reality with

the closest distance to the target

• Advantanges
• robuster network
• faster search

24

More Realities

• Advantages
• robuster
• shorter paths

25

• Dimensionens

reduce the lookup path length more effciently

• Realities

produce more robust networks

26

Realities vs. Dimensions

Peer-to-Peer Networks

03 CAN (Content Addressable Network)

Christian Ortolf

Technical Faculty Computer-Networks and Telematics University of Freiburg